Simplify; express your answer in exponential form. Assume $q\neq 0, r\neq 0$. $\dfrac{{(q^{3}r^{-5})^{3}}}{{(q^{-5}r^{2})^{3}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{3}r^{-5})^{3} = (q^{3})^{3}(r^{-5})^{3}}$ On the left, we have ${q^{3}}$ to the exponent ${3}$ . Now ${3 \times 3 = 9}$ , so ${(q^{3})^{3} = q^{9}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{3}r^{-5})^{3}}}{{(q^{-5}r^{2})^{3}}} = \dfrac{{q^{9}r^{-15}}}{{q^{-15}r^{6}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{9}r^{-15}}}{{q^{-15}r^{6}}} = \dfrac{{q^{9}}}{{q^{-15}}} \cdot \dfrac{{r^{-15}}}{{r^{6}}} = q^{{9} - {(-15)}} \cdot r^{{-15} - {6}} = q^{24}r^{-21}$